We briefly describe the procedure to remove all the higher-order time derivatives,
using coordinate transformations [1]}, [2]}, [3]}, [4]}, [5]}.
The Lagrangian under a coordinate transformation \(\textbf {x}_{(a)} \rightarrow \textbf {x}_{(a)} + \delta \textbf {x}_{(a)}\)  changes
by
\(\delta \mathcal {L} = \left( \frac{\delta \mathcal {L}}{\delta \textbf {x}_{(a)}^i} \right)~ \delta \textbf {x}_{(a)}^i + \mathcal {O}\left(\delta \textbf {x}_{(a)}^2\right)\) 
